Conjugate reversibility in complex special linear groups

TL;DR

This paper introduces and classifies conjugate reversibility in complex special linear groups, providing algebraic characterizations and detailed classifications, especially for $ ext{SL}(4, ext{C})$, with implications for projective transformations.

Contribution

It defines conjugate reversibility in $ ext{SL}(n, ext{C})$, proves that all such elements are strongly conjugate reversible, and offers a complete classification based on conjugacy invariants.

Findings

Every $c$-reversible element is strongly $c$-reversible in $ ext{SL}(n, ext{C})

Complete classification of $c$-reversible elements via conjugacy invariants

Finer classification in $ ext{SL}(4, ext{C})$ using trace conditions and resultants

Abstract

We introduce and study conjugate reversibility (or $c$-reversibility) in the complex special linear group $\SL(n,\C)$ where an element is conjugate to the inverse of its complex conjugate. We prove that in $\SL(n, \C)$, every $c$-reversible element is strongly $c$-reversible. We provide a complete classification of $c$-reversible elements based on their conjugacy invariants. This leads to an algebraic characterization of projective transformations. As a special case, a finer classification in $\SL(4, \C)$ is obtained in terms of trace conditions and resultant computations.